Vo Thanh Liem

An α-approximation theorem for Q∞-manifolds

Abstract Given a Q ∞ -manifold M and an open cover α of M , there is an open cover β of M such that every β -equivalence from a Q ∞ -manifold N to M is α-close to a homeomorphism. Consequently, every Q ∞ -deficient subset in a C ∞ -manifold M is strongly negligible in M .

On the asphericity of knot complements

Two examples of topological embeddings of S 2 in S 4 are constructed. The first has the unusual property that the fundamental group of the complement is isomorphic to the integers while the second homotopy group of the complement is non-trivial. The second example is a non-locally flat embedding whose complement exhibits this property locally. Two theorems are proved. The first answers the question of just when good π 1 implies the vanishing of the higher homotopy groups for knot complements in 54. The second theorem characterizes local flatness for 2-spheres in S 4 in terms of a local π 1 condition

Manifolds accepting codimension-one- sphere-shape decompositions

Abstract A decomposition of a metric space is said to be CS k -shape if each of its members is a compactum shape equivalent to a cohomology k -sphere. We will show that for m ⩾2 every CS m −1 -shape decomposition of a closed m -manifold is upper semicontinuous (Theorem 3.1). Consequently, for m ≠3, 4, 5, every connected closed m -manifold accepting an S m −1 -shape decomposition is homeomorphic to the total space of an ( m −1)-sphere-fiber bundle over the circle (Theorem 4.2).

A HOMOTOPY EQUIVALENCE THAT IS NOT HOMOTOPIC TO A TOPOLOGICAL EMBEDDING

A b st r a c t . An open subset W of S n , n ‚ 6, and a homotopy equivalence f : S 2 ◊S ni4 ! W are constructed having the property that f is not homotopic to any topological embedding.

Characterization of knot complements in the 4-sphere☆☆☆

Abstract Knot complements in S4 are characterized as follows: A connected open set W ⊂ S4 is homeomorphic to the complement of some locally flat 2-sphere in S4 if and only if H1(W) is infinite cyclic, W has one end, and the fundamental group of that end is infinite cyclic. Applications include a characterization of weakly flat 2-spheres in S4 and a complement theorem for 2-spheres in S4.

Representing homology classes of simply connected 4-manifolds

Abstract The main theorem asserts that every 2-dimensional homology class of a compact simply connected PL 4-manifold can be represented by a codimension-0 submanifold consisting of a contractible manifold with a single 2-handle attached. One consequence of the theorem is the fact that every map of S2 into a simply connected, compact PL 4-manifold is homotopic to an embedding if and only if the same is true for every homotopy equivalence. The theorem is also the main ingredient in the proof of the following result: If W is a compact, simply connected, PL submanifold of S4, then each element of H 2 (W; Z ) can be represented by a locally flat topological embedding of S2.